Is there any theory that proves that amphibians have evolved from fish

what are the connections between the vertebrate hearts?are there any theories or a man who discovered it?

Here's some of the basics on tetrapod evolution:

http://evolution.berkeley.edu/evolibrary…
http://en.wikipedia.org/wiki/Tetrapod

And a comparison of hearts specifically:

http://users.rcn.com/jkimball.ma.ultrane…

I think you'll find your answer if you look at two different species.

First of all you have the mudskipper, which is a fish that spends a significant amount of time on land on muddy flats and can breath both through its gills and through its skin. But it doesn't have legs, it still has fins with which it 'walks' across the mud flats it inhabits. You can imagine that millions of year ago, modern amphibians would have developed from some similar animal, whose fins developed and became more and more adapted to the land until they became what we'd call legs.
http://www.youtube.com/watch?v=KurTiX4FD…

Then you have the mudpuppy, a salamander that due to its predominantly aquatic environment and maintains many fishlike qualities such as gills.

Theories do not prove they explain. Evidence is what supports a theory, or disproves it, are you perhaps asking if there is evidence of amphibian evolution?

You may want to read these links;
http://rationalwiki.org/wiki/Tiktaalik
http://tiktaalik.uchicago.edu/
http://scienceblogs.com/pharyngula/2006/…
http://www.zsl.org/zsl-london-zoo/exhibi…
http://dinosaurs.about.com/od/otherprehi…
http://en.wikipedia.org/wiki/Evidence_of…

a theory cant prove stuff i think the word your looking for is fact XD